Mock 5 – Mathematical Logic (Part 3)

Which of the following statements are true if the universe of discourse is the set of integers? (A) ∀n (n² ≥ 0) (B) ∃n (n² = 2) (C) ∀n (n² ≥ n) (D) ∃n (n² < 0)

Match List - I with List - II.

The negation of "Some students like hockey" is:

Considering α, β, γ as logical variables, which represent correct equivalences? (A) (α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨ (α ∧ γ)) (B) (α ∨ β) ≡ ¬α ∨ β (C) (α ⇒ β) ≡ (¬β ⇒ ¬α) (D) ¬(α ∨ β) ≡ (¬α ⇒ ¬β)

Assertion (A): p̄. Reason (R): (r → q̄, r ∨ s, s → q̄, p → q). Choose the correct answer.

Which of the following are logically equivalent? A. ¬p → (q → r) and q → (p ∨ r) B. (p → q) → r and p → (q → r) C. (p → q) → (r → s) and (p → r) → (q → s)

Statement I: ⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋ for every real x. Statement II: ⌈x + y⌉ = ⌈x⌉ + ⌈y⌉ for all real x and y.

Let (X, *) be a semigroup in which, for every a, b in X, a ≠ b implies a * b ≠ b * a. Which equalities hold? A. a * a = a for every a. B. a * b * a = a for all a, b. C. a * b * c = a * c for all a, b, c.

Which of the following pairs of propositions are not logically equivalent?

If f(x)=x is my friend, and p(x)=x is perfect, then the correct logical translation of the statement "some of my friends are not perfect" is:

Consider the argument with premise ∀x(P(x) ∨ Q(x)) and conclusion (∀x P(x)) ∧ (∀x Q(x)): (A) ∀x(P(x) ∨ Q(x)) premise; (B) P(c) ∨ Q(c) universal instantiation from A; (C) P(c) simplification from B; (D) ∀x P(x) universal generalization of C; (E) Q(c) simplification from B; (F) ∀x Q(x) universal generalization of E; (G) (∀x P(x)) ∧ (∀x Q(x)) conjunction of D and F.

Consider the statement: A person who is radical (R) is electable (E) if he/she is conservative (C), but otherwise is not electable. Which of these assertions are correct? (A) (R ∧ E) ⇔ C (B) R ⇒ (E ⇔ C) (C) R ⇒ ((C ⇒ E) ∨ ¬E) (D) (¬R ∨ ¬E ∨ C) ∧ (¬R ∨ ¬C ∨ E)

Consider the following statements:

S : ∀xP(x) ∨ ∀xQ(x) 1 and ∀x(P(x) ∨ Q(x)) are not logically equivalent.

S : ∃xP(x) ∧ ∃xQ(x) 2 and ∃x(P(x) ∧ Q(x)) are not logically equivalent Which of the following statements is/are correct?

Which of the following is the principal conjunctive normal form of [(p ∨ q) ∧ ¬p] → ¬q?

Match List I with List II. List I: (a) p → q, (b) p ∨ q, (c) p ∧ q, (d) ¬(p → q). List II: (i) ¬(q → ¬p), (ii) p ∧ ¬q, (iii) ¬p → q, (iv) ¬p ∨ q.

In mathematical logic, which of the following are statements?

(i) There will be snow in January.

(ii) What is the time now ?

(iii) Today is Sunday.

(iv) You must study Discrete Mathematics.

Choose the correct answer from the code given below :

Match List I with List II and choose the correct answer from the code given below.

Digital data from a sensor can fill 0 to 32 buffers. Let the sample space be S = {0, 1, 2, …, 32}, where j means j buffers are full, and p(i) = (33 − i)/561. Let A be the event that an even number of buffers are full. Then p(A) is

The equivalence of ¬∃x Q(x) is